Existence of generalized solutions for Keller-Segel-Navier-Stokes equations with degradation in dimension three

1.   Introduction

We consider a mathematical model to decribe the dynamics of biological organism influenced by chemical signal and living in fluid. The original Keller-Segel system was proposed to write the motion of biological individuals sensing gradient of a chemical substance and moving toward its higher concentration (see ^[9]). Such biological organisms often live in fluid, and thus their behaviors are influenced by motions of viscous fluid flows as well. There are, for example, the bacteria living in fluid such as Bacillus subtilus (^{[1,2,7,11,18,24]}) or Escherichia coli (^[12,22]) or phenomena of coral fertilization in sea resulting from the chemotatic behavior of sperm (^[4,6,10,24]).

In this note, we study the following Keller-Segel system with degradation coupled to the Navier-Stokes equations in a bounded domain in three dimensions:

in $\Omega\times (0, T)$, where $\Omega\subset \mathbb{R}^3$ is a bounded domain with smooth boundary and $T > 0$. Here $n$, $c$, $u$, and $P$ are the population density of the chemotactic organisms, the concentration of signal substances, the fluid velocity, and the associated pressure, respectively. No flux condition is assigned for $n$ and $c$ at the boundary, and $u$ has no slip boundary condition there, namely

We assume that initial data $(n_0, c_0, u_0)$ satisfies

In case that the Eq (1.1) has the logistic degradation, i.e., $q = 2$, Tao and Winkler ^[16] proved global existence and large time behavior of classical solutions to the system (1.1)–(1.3) in two dimensions. Such result was extended to the case of three dimensions, provided that the fluid equation is given by the Stokes system, instead of the Navier-Stokes equations, and $\mu$ is sufficiently large (see ^[15]).

For the chemotaxis-Navier-Stokes system (1.1)–(1.3) with $q = 2$, the existence of generalized solutions was proved by Winkler ^[22].

To the best of our knowledge, if $q < 2$, it is not known whether or not classical solutions exist globally in time for general data and parameters. Instead of classical solutions, recently it was shown in ^[8] that generalized solutions to the chemotaxis-Stokes system exists globally in time for $q\in (2-\frac{1}{d}, 2)$, where $d$ is dimensions two or three, i.e., $d = 2, 3$. (the notion of generalized solutions is reminded in Definition 2). In the absence of fluid, i.e., $u = 0$, one can refer to ^[19,20,23] for generalized solutions.

The main objective of this note is to establish the existence of generalized solutions globally in time, in case that the degradation power $q$ is less than 2, and the Navier-Stokes equations are coupled for the fluid equations in three dimensions.

To begin with, we recall the notion of generalized solution of (1.1)–(1.3). Firstly, we remind the $\gamma-$entrophy super(or sub) solution of the Eq (1.1).

Definition 1. Let ${\gamma}\in(0, 1)$. Assume that a pair of functions $(n, c)$ and a vector field $u$ satisfy the following:

Then such $(n, c, u)$ is called a weak ${\gamma}$-entropy super-solution(resp., sub-) of the first equation in (1.1)–(1.3) if

for all nonnegative ${\varphi}\in C^\infty_0(\overline{\Omega}\times[0, \infty))$.

Next, we define the notion of the generalized solutions of (1.1)–(1.3).

Definition 2. A triple of two functions and a vector field

satisfying

is called a generalized solution of (1.1)–(1.3), if

for all ${\varphi}\in C^\infty_0(\overline{ {\Omega}}\times [0, \infty))$ and, if ${{\nabla }\cdot}{u} = 0$ in $\mathcal{D}'(\Omega\times(0, \infty))$ and

for all ${\varphi}\in C^\infty_0({\Omega}\times [0, \infty); \mathbb{R}^3)$ with ${{\nabla }\cdot} {\varphi}\equiv 0$, and if there exist ${\gamma}_1, {\gamma}_2\in(0, 1)$ such that $(n, c, u)$ is a weak ${\gamma}_1$-entropy super-solution and a weak ${\gamma}_2$-entropy sub-solution of the first equations in (1.1)–(1.3).

For logistic coefficients $\rho, \mu$ and the potential function $\phi$, we assume that

We are now ready to state our main result.

Theorem 1.1. Let $q\in\left({\frac{20}{11}, 2} \right)$. Then the Eqs $(1.1)–(1.5)$ with $(1.8)$ admit at least one generalized solution in the sense of Definition 2.

Remark 1. The result Theorem 1.1 is an improvemnt of that of ^[22], which showed the existence of the generalized solution in case that $q = 2$. Furthermore, it is also an extension to the result of ^[8], since the Navier-Stokes equations are considered instead the Stokes system. In such case, the range of $q$ is, however, restrictive, compared to the case that $q\in\left({\frac{5}{3}, 2} \right)$ in ^[8]. This is mainly due to the fact that the control of $u$ is more difficult for the Navier-Stokes equations, which causes lower regularity of $u\cdot\nabla c$ and, in turn, $\nabla c$ (see Lemma 3.6 for the details). Therefore, passing to the limit for regularized solutions, convergence to $n\nabla c$ is well understood only for $q\in \left({\frac{20}{11}, 2} \right)$.

Next, in case that $\rho \leqslant0$, we can show that generalized solutions converge to zero in an appropriate sense, passing time to the limit. More precisely, we obtain the following:

Theorem 1.2. Let $(n, c, u)$ be the generalized solution established in Theorem 1.1. If $\rho = 0$, then $(n, c, u)$ vanishes in $L^1(\Omega)\times L^l(\Omega)\times L^2(\Omega)$ as time tends to infinity. Furthermore, $(n, c, u)$ satisfies

Morerover, if $\rho < 0$, then $(n, c, u)$ satisfies

where $\rho_* = \min\left\{ {-\rho, 1} \right\}$, $\delta_* = \frac{1}{2}\min\left\{ {\frac{C_p}{2}, -\rho\frac{5q-6}{3(q-1)}} \right\}$ and $C_p$ is the Poincaré constant for $\Omega$.

Remark 2. The result of Theorem 1.2 can be extended to the case $q = 2$ and $\rho = 0$. In such case, in particular, estimates of $c$ read as follows:

This estimate of decay for $c$ is slightly better, compared to those of [22,Section 8]. On the other hand, in case that $q = 2$ and $\rho > 0$, it was also shown in ^[22] that if $\mu > \chi\sqrt{\rho}/4$, then

This convergence is based on stabilization of a certain energy functional (see [22,Section 8]). Although similar results are expected, such a method doesn't seem to be valid unless $q = 2$. Therefore, we leave the asymptotic behaviors as an open question in case that $\rho > 0$ and $q < 2$.

This paper is organized as follows: In Section 2, we introduce an approximated system and recall some useful lemma for our purpose. Section 3 is devoted to obtaining estimates, independent of a regularizing parameter, of the approximated system. We then discuss the convergence of approximated solutions to a generalized solution in Section 4. Finally, in Section 5, asymptotic estimates are provided.

Throughout this paper, we shall abbreviate $\left\Vert {f} \right\Vert_{L^p({\Omega})}$ as $\left\Vert {f} \right\Vert_p$ for simplicity. Further, we denote by $C>0$ generic constants which may be different from line to line.

2.   Preliminaries

In the following proposition we define an appropriate approximated system to (1.1)–(1.3), for which global classical solutions can be verified. The approximated system is given by

Here ${\epsilon}\in(0, 1)$, ${\kappa}>2$ and $Y_ {\epsilon}$ is the Yosida approximation defined by

where A is the realization of the stokes operator in $D(A) = W^{2, 2}({\Omega})\cap W^{1, 2}_{0, \sigma}({\Omega})\subset L^2_\sigma({\Omega})$.

Following method of proofs developped in ^[8] and ^[22], one can prove the existence of classical solution of the approximated system (2.1). Since its verification is similar to thoes of ^[8] and ^[22], we skip its proof.

Proposition 1. For each ${\epsilon}\in(0, 1)$, there exist functions

such that $(n_ {\epsilon}, c_ {\epsilon}, u_ {\epsilon}, P_ {\epsilon})$ solves $(2.1)$ classically in $\overline{ {\Omega}}\times (0, \infty)$.

We recall an effective inequality in Sobolev spaces called the Gagliardo-Nirenberg interpolation inequality. Here we only consider a version of bounded Lipschitz domain $\Omega$ in $\mathbb{R}^3$. The proof can be found in [3,Theorem 1.5.2] and ^[13].

Lemma 2.1. Let $1 \leqslant p, r \leqslant \infty$ and $0 \leqslant n < m\in\mathbb{N}$. Then there exist constants $C_1$ and $C_2>0$ such that

where $\frac{1}{q}-\frac{n}{3} = \left({\frac{1}{p}-\frac{m}{3}} \right)\theta+\frac{1}{r}(1-\theta)$, $\theta\in [\frac{n}{m}, 1]$, and $s>0$ is arbitrary.

The following two Lemmas named maximal estimates are crutial to obtain a regularity of approximated solutions (see ^[5,8,14]).

Lemma 2.2. Let $T>0$, $v_0\in W^{1, p}(\Omega)$ and $h\in L^p(0, T;L^p(\Omega; \mathbb{R}^3))$ for $1 < p < \infty$. Then there exists a unique solution $v\in L^p(0, T;W^{1, p}(\Omega))$ solving

Furthermore, $v$ attains the following estimate.

Lemma 2.3. Let $T>0$ and $p\in(1, 2]$. Then for every $v_0\in W^{1, \infty}({\Omega})$ and $h\in L^p\left({ {\Omega}\times(0, T)} \right)$, the following heat equation with Neumann boundary condition

has a unique solution $v\in W^{1, p}((0, T);L^p({\Omega}))\cap L^p((0, T);W^{2, p}({\Omega}))$ satisfying

with some $C_T>0$.

Proof. Set $X = L^p({\Omega})$ and $X_1 = W^{2, p}_\nu({\Omega}): = \{f\in W^{2, p}({\Omega}) : \frac{\partial {f}}{\partial {\nu}} = 0$ on ${\partial} {\Omega}\}$. From ^[14] and [19,Proposition 2] we have

where $\left\Vert {\cdot} \right\Vert_{1-\frac{1}{p}, p}$ stands for the norm in the real interpolation space $\left({X, X_1} \right)_{1-\frac{1}{p}, p}$. Now (2.5) is achieved from the embedding [21,Lemma 2.1.(ii)]

for any $p\in(1, 2]$.

Remark 3. For the purpose of our analysis, we consider only the case $p\in(1, 2]$ in Lemma 2.3. One can refer to ^[21] for more general cases, in particular $p \geqslant3$, where the interpolation space $\left({X, X_1} \right)_{1-\frac{1}{p}, p}$ is not equaivalent to $W^{2(1-\frac{1}{p}), p}({\Omega})$.

Next, we present a compactness theorem called Aubin-Lions Lemma [17,Theorem 2.1] that will be used to give convergence results for the approximated solution $(n_ {\epsilon}, c_ {\epsilon}, u_ {\epsilon})$.

Lemma 2.4. Let $T>0$, $1 \leqslant {\alpha}_0, {\alpha}_1 < \infty$ and $X_0$, $X$, $X_1$ be Banach spaces with $X_0\subset X\subset X_1$. Suppose further that the embedding $X_0\hookrightarrow X$ is compact and the embedding $X\hookrightarrow X_1$ is continuous. Let

Then the embedding $W\hookrightarrow L^{ {\alpha}_0}(0, T; X)$ is compact.

3.   Regularized solutions

The following basic properties of these solutions are well-known.

Lemma 3.1. Let $T>0$. For each ${\epsilon}\in(0, 1)$, the solution of $(2.1)$ fulfills

and

where $m = \max\left\{ {{\int\limits_{ {\Omega}}n_0, \left({\frac{\rho_{+}\left| { {\Omega}} \right|}{\mu}} \right)^\frac{1}{q-1}}} \right\}$ and $\rho_+ = \max\left\{ {\rho, 0} \right\}$.

Proof. Integrating the first equation in (2.1) over ${\Omega}$, employing the divergence theorem, and using the Hölder inequality yield that, for all $t>0$,

An ODE comparison implies (3.1). Integrating (3.3) with respect to time and then using (3.1), we have

which implies (3.2).

The following estimate is easily obtained by (3.1).

Lemma 3.2. For each ${\epsilon}\in(0, 1)$, we have

Proof. Integrating the equation for $c_ {\epsilon}$ in (2.1) and using (3.1), we have

which yields (3.4) by the ODE comparison.

We recall a useful result shown in [22,Lemma 3.4].

Lemma 3.3. Let $T\in(0, \infty]$, $\tau\in(0, T)$, $a>0$ and $b>0$. Suppose that a nonnegative function $h\in L^1_{loc}(\mathbb{R})$ be such that

If a nonnegative function $y\in C^0[0, T)\cap C^1(0, T)$ satisfies

then

The following lemma is a variant of the result with $q = 2$ in [22,Lemma 3.6].

Lemma 3.4. Let $T>0$ and $q\in(\frac{5}{3}, 2)$. Then there exists $C>0$ such that for any ${\epsilon}\in(0, 1)$ we obtain

Moreover,

where $r = \frac{3q}{5-2q}$.

Proof. Multiplying the equation for $c_ {\epsilon}$ in (2.1) by $c_ {\epsilon}^{r-1}$ and integrating over $\Omega$, we have for all $t>0$,

where the Hölder inequality is used. Using the Gagliardo-Nirenberg inequality and (3.4), we note that

where $\theta = \frac{3}{2}(1-\frac{r}{r-1}\frac{g-1}{q})\in(0, 1)$ since $r = \frac{3q}{5-2q}$. Employing Young's inequality, we have

Combining (3.7) with (3.8) implies that there exist $C_5>0$ such that

Let $y(t): = \left\Vert {c_ {\epsilon}(t)} \right\Vert^r_r+1$ and $h(t): = \left\Vert {n_ {\epsilon}(t)} \right\Vert^q_q+1$, which is in $L^1$ locally in time. Then, dividing (3.9) by $y(t)$ yields that

We use again the Gagliardo-Nirenberg inequality to obtain that for all $t>0$

which leads that $\left\Vert { {\nabla } c_ {\epsilon}^\frac{r}{2}} \right\Vert_2^2 \geqslant\left({\frac{1}{C}y(t)-1} \right)^\frac{3r-1}{3(r-1)} \geqslant Cy^\frac{3r-1}{3(r-1)}-1$. Hence, it follows that

where we use the trivial inequality $\ln{y}\lesssim y^k$ for $k>0$. Putting the above inequality (3.11) into (3.10), we have

By Lemma 3.3, we can conclude that there exists $C>0$ satisfying $y(t) \leqslant C$ for all $t>0$ which proves (3.5) as required. Integrating (3.10) with respect to time and exploiting the boundedness of $y(t)$, guaranteed by (3.5), yield that

for some $C>0$. Using (2.2) and (3.4), we finally have (3.6).

We adopt well-known energy estimate for the Navier-Stokes system to gain a bound for $u_ {\epsilon}$ in energy class.

Lemma 3.5. Let $T>0$ and $q\in\left({\frac{5}{3}, 2} \right)$. Then there exists $C>0$ such that for each ${\epsilon}\in(0, 1)$, we have

and

Proof. We test the fluid equation in (2.1) by $u_ {\epsilon}$ to find the following $L^2$ estimate

We can estimate the right hand side of (3.14) using the H$\ddot{\rm o}$lder inequality, the Sobolev embedding $W^{1, 2}_{0, \sigma}\hookrightarrow L^6$, and the interpolation inequality for $n_ {\epsilon}$ that

where we used that $\frac{q}{3(q-1)} \leqslant q$.

Thus, with the aid of (3.15) and the Poincaré inequality, we have for some $C$

(3.12) is proved if we use (3.2) and Lemma 3.3, and then (3.13) can be calculated by integrating (3.14) with respect to time and using (3.15).

A direct consequence of Lemma 3.5 is the following.

Corollary 1. Let $T>0$ and $\frac{3}{ {\alpha}}+\frac{2}{ {\beta}} = \frac{3}{2}$, $2 \leqslant {\alpha} \leqslant6$. Then

in particular, if ${\alpha} = {\beta} = \frac{10}{3}$, then

Proof. In view of Lemma 3.5, (3.16), in particular (3.17), is derived from the Gagliardo-Nirenberg inequality (2.2).

Since $u_{\epsilon}$ only belong to energy class, we have lower regularity of ${\nabla } c_{ {\epsilon}}$, due to difficulties of controlling convective term $u\cdot {\nabla } c$, than the case that the Stokes sysem is coupled. Nevertheless, using the divergence free condtion, we obtain a certain integrability of ${\nabla } c_{ {\epsilon}}$ by the following decompsition, which makes computations easier. More precisely, let $w_ {\epsilon}$ be a solution satisfying

Now we set $\tilde{w}_ {\epsilon}: = c_ {\epsilon}-w_ {\epsilon}$. Then, due to the divergence free condition for $u_ {\epsilon}$, it follows that $\tilde{w}_ {\epsilon}$ solves

In next lemma, estimating each solutions of the decompsition, we show that $\nabla c_{\epsilon}\in L^{10q/(10-q)}(\Omega\times (0, T))$.

Lemma 3.6. Let $T>0$ and $q\in\left({\frac{5}{3}, 2} \right)$. Then given ${\epsilon}\in(0, 1)$, there exists $C = C(T)>0$ such that

where $m = \frac{10q}{10-q}$.

Proof. We first observe reularity of $w_ {\epsilon}$. On account of (2.5), we can find a constant $C = C(T)>0$ satisfying

Then the Gagliardo-Nirenberg interpolation inequality (2.2) and (3.5) yield that

Thus, from (3.19) and (3.20) we see that for some $C = C(T)>0$

The last term is finite because of (3.2), (3.5) and the fact that $q \leqslant r = \frac{3q}{5-2q}$. Next, let ${\alpha}$ and ${\beta}$ be in Lemma 3.5 with ${\alpha} = \frac{90q}{11q+40}$ and ${\beta} = \frac{30q}{17q-20}$. It can be easily checked that $2 < {\alpha} < 6$ and $2 < {\beta}$ because $q\in\left({\frac{5}{3}, 2} \right)$. Then we can see via the maximal estimate (2.3) and the Hölder inequality that

which is valid since $\frac{1}{m} = \frac{1}{ {\alpha}}+\frac{1}{3r} = \frac{1}{ {\beta}}+\frac{1}{r}$, where $r = \frac{3q}{5-2q}$. The last term in (3.21) is finite due to (3.16) and (3.6). Hence, we have

which is finite since $m < \frac{5q}{5-q}$ and (3.21). Then (3.18) is proved.

Taking advantage of Lemma 3.6, we can obtain the maximal estimate for $c_ {\epsilon}$.

Lemma 3.7. Let $T>0$ and $q\in\left({\frac{5}{3}, 2} \right)$. Then there exists $C = C(T)>0$ such that for any ${\epsilon}>0$,

Proof. Applying (2.5), we obtain

due to (3.2), (3.5), (3.17) and (3.18). This proves (3.22).

The following two lemmas are crucial to achieving the convergence property for $n_ {\epsilon}$.

Lemma 3.8. Let $T>0$ and $q\in\left({\frac{5}{3}, 2} \right)$. Then for any ${\gamma}\in(0, 1)$ with ${\gamma} \leqslant\frac{4q-5}{5}$, there exists $C = C(T)>0$ satisfying

Proof. Testing the first equation in (2.1) by ${\gamma} n_ {\epsilon}^{ {\gamma}-1}$ and using integration by parts, we obtain

Using Young's inequality and (3.2), we have

and

Since $0 < {\gamma} \leqslant\frac{4q-5}{5}$, we see that $\frac{5+q}{5q}+\frac{ {\gamma}}{q} \leqslant1$. This leads

Collecting (3.24), (3.25) and (3.26), we obtain

Since ${\gamma}-2 < 0$, we get $(n_ {\epsilon}+1)^{ {\gamma}-2} \leqslant n_ {\epsilon}^{ {\gamma}-2}$, hence (3.23).

In the following lemma, we mean by $(W^{k, 2}_0)^*$ the dual space of $W^{k, 2}_0$.

Lemma 3.9. Let $T>0$ and $q\in\left({\frac{5}{3}, 2} \right)$. Then for any ${\gamma}\in(0, 1)$ with ${\gamma} \leqslant\frac{4q-5}{5}$, there exists $k\in\mathbb{N}$ and $C = C(T)>0$, independent of ${\epsilon}$, satisfying

Proof. Fix $k\in\mathbb{N}$ to be choosen later and let ${\varphi}\in W^{k, 2}_0(\Omega)$ be a test function. We observe that

First, employing integration by parts and H$\ddot{\rm o}$lder inequality, we can estimate $J_1$ as follows:

where we used the fact that $(1+n_ {\epsilon})^{\frac{ {\gamma}}{2}-2} \leqslant (1+n_ {\epsilon})^{ {\gamma}-2} \leqslant n_ {\epsilon}^{ {\gamma}-2}$. Similarly, the second and third terms are controlled as follows:

because ${\gamma} < 1 < \frac{7q}{5}$ and $\frac{10q}{7q-5 {\gamma}} \leqslant\frac{10q}{3q+5}$.

where we used the fact that $q < m$ and ${\gamma} \leqslant\frac{4q-5}{5} < 2q-2$. Estimates for $J_4, J_5$ and $J_6$ can be easily obtained by the following calculation

Collecting all of estimates (3.28)-(3.33) and applying the Sobolev embedding theorem, we have

Choose $k$ sufficiently large that $k>\frac{5}{2}$. Then $W^{k, 2}_0(\Omega)$ is embedded into $W^{1, \infty}(\Omega)$ by Sobolev embedding. Finally, integration of (3.34) over $(0, T)$ leads, with the help of (3.1), (3.2), (3.18), (3.16) and (3.23), that

as desired.

The estimate for the time derivative of $u_ {\epsilon}$ is obtained by the simple calculation.

Lemma 3.10. Let $T>0$. Then there exists $C>0$ such that for any ${\epsilon}>0$,

Proof. Given ${\varphi}\in C^\infty_0({\Omega}\times [0, \infty); \mathbb{R}^3)$ with ${{\nabla }\cdot} {\varphi} = 0$, we compute

Here we used the well-known inequality $\left\Vert {Y_ {\epsilon} u_ {\epsilon}} \right\Vert_2^2 \leqslant C\left\Vert {u_ {\epsilon}} \right\Vert_2^2$. Thus, integrating (3.36) over $(0, T)$ yields (3.35).

4.   Convergence

We are now ready to prove the convergence property for $(n_ {\epsilon}, c_ {\epsilon}, u_ {\epsilon})$.

Lemma 4.1. Let $q\in(\frac{5}{3}, 2)$, ${\gamma}\in(0, 1)$ with ${\gamma} \leqslant\frac{4q-5}{5}$ and $p\in(1, q)$. A number $m$ is given in Lemma 3.6. Then the classical solution $(n_ {\epsilon}, c_ {\epsilon}, u_ {\epsilon})$ of (2.1) satisfies the following convergence property.

Proof. For convenience, we denote a subsequence $({\epsilon}_j)_{j\in\mathbb{N}}$ of ${\epsilon}$ by ${\epsilon}$ itself. First, Lemma 2.4 gives the pointwise convergence of $c_ {\epsilon}$ in (4.5):

Indeed, using Lemma 2.4, bounds for $c_ {\epsilon}$ in $L^m_{loc}([0, \infty); W^{1, m}(\Omega))$ and ${\partial}_t c_ {\epsilon}$ in $L^\frac{5q}{5+q}_{loc}(\overline{\Omega}\times[0, \infty))$, asserted in Lemma 3.6 and Lemma 3.7, yield the strong convergence of $c_ {\epsilon}$ in $L^m_{loc}(\overline{\Omega}\times[0, \infty))$ which in particular implies (4.5). Similarly, by Lemma 3.8 and 3.9, we see that $\left({1+n_ {\epsilon}} \right)^\frac{ {\gamma}}{2}_{ {\epsilon}\in(0, 1)}$ is relatively compact in $L^2_{loc}(\overline{\Omega}\times[0, \infty))$ with respect to the strong topology by Lemma 2.4. we can thus see that

which proves (4.1), as well as (4.4) holds. Likewise, exploiting boundedness of $u_ {\epsilon}$ and of its time derivative, as proved in Lemma 3.5 and Lemma 3.10, and using Lemma 2.4 again, we have (4.8) and (4.9). The convergence properites (4.2), (4.6), (4.7), (4.10) and (4.11) is a direct consequence of (3.2), (3.18), (3.22), (3.17) and (3.13), respectively. In order to prove (4.3), we use (3.2) again, which implies that $\int_{0}^{T}\left\Vert {n_ {\epsilon}^p} \right\Vert_\frac{q}{p} \leqslant C$ for all $t>0$. Hence we have

as ${\epsilon}\searrow 0$. By this weak convergence we have

which asserts that $n_ {\epsilon}\rightarrow n \text{ in }L^p_{loc}(\overline{\Omega}\times[0, \infty))$ due to uniform convexity of $L^p$-space for $p>1$. This proves (4.3).

We shall prove the limit $(n, c, u)$ in Lemma 4.1 is a solution of our main system (1.1)–(1.3) in the sense of Definition 2. We first focus on $c$ and $u$ which satisfy (1.1) and (1.2) in the standard weak sence. In addition, we show that $n$ is a weak sub-solution in the sense of Definition 1.

Lemma 4.2. Let $(n, c, u)$ be the limit function and vector field in Lemma 4.1. Then $(1.6)$ and $(1.7)$ hold.

Proof. We multiply the second equation in (2.1) by the test function ${\varphi}\in C^\infty_0(\overline{\Omega}\times[0, \infty))$ to get, for all ${\epsilon}\in(0, 1)$,

Applying (4.6) and (4.2), we easily obtain

as ${\epsilon} = {\epsilon}_j\searrow 0$. On the other hand, combining (4.3) and (4.10) infers that $c_ {\epsilon} u_ {\epsilon}\rightharpoonup cu$ in $L^s_{loc}$ for $s: = \frac{10+3p}{10p} \geqslant1$, which proves

as ${\epsilon}\searrow 0$. Next we multiply the third equation in (2.1) by ${\varphi}\in C^\infty_0({\Omega}\times [0, \infty); \mathbb{R}^3)$ with ${{\nabla }\cdot} {\varphi} = 0$ that gives

for all ${\epsilon}\in(0, 1)$. Similar to the above, (4.10), (4.11), (4.2) and the condition on ${\nabla }\phi$, as assumed in (1.8), imply that

as ${\epsilon}\searrow 0$. Since it is well known that $Y_ {\epsilon} u_ {\epsilon}\, \rightarrow u$ in $L^2_{loc}({\Omega}\times(0, \infty))$, with the aid of (4.9), we obtain $Y_ {\epsilon} u_ {\epsilon}\otimes u_ {\epsilon}\rightarrow u\otimes u$ in $L^1_{loc}({\Omega}\times(0, \infty)).$ This proves

as ${\epsilon}\searrow 0$. We collect (4.12)–(4.17) to conclude the proof.

So far, we used that $q>\frac{5}{3}$. In the next Lemma, however, it is necessary to assume that $q>\frac{20}{11}$, which is crucial to show convergence of $n_ {\epsilon} {\nabla } c_ {\epsilon}$ (see the estimate (4.21) below).

Lemma 4.3. Let $q\in\left({\frac{20}{11}, 2} \right)$ and $(n, c, u)$ be the limit function and vector field in Lemma 4.1. Then $n$ is a ${\gamma}-$entropy sub-solution of $(1.1)–(1.3)$ with ${\gamma} = 1$, that is, $n$ satisfies the following integral inequality

for all nonnegative ${\varphi}\in C^\infty_0(\overline{ {\Omega}}\times[0, \infty))$.

Proof. We multiply the first equation in (2.1) by a nonnegative test function ${\varphi}\in C^\infty_0 (\overline{ {\Omega}}\times[0, \infty))$ and integrate over ${\Omega}\times(0, \infty)$. By suitable integration by parts,

for all ${\epsilon}\in(0, 1)$. Using (4.2), we see that

as ${\epsilon}\searrow 0$. Funthermore, applying strong convergence of $(n_ {\epsilon})_{ {\epsilon}\in(0, 1)}$, $(u_ {\epsilon})_{ {\epsilon}\in(0, 1)}$ as asserted in Lemma 4.1, we have

as ${\epsilon}\searrow 0.$ Since $q\in(\frac{20}{11}, 2)$, we can take $p < q$ close to $q$ satisfying $\frac{1}{p}+\frac{1}{m} < 1$. Then, by (4.3) and (4.6) we see that

as ${\epsilon}\searrow 0$. Besides, the nonnegativity of $n_ {\epsilon}$ and ${\varphi}$ leads that

for all ${\epsilon}\in(0, 1)$. Lastly, we observe that by Fatou's lemma

Hence, combining (4.18)–(4.23), we conclude that $n$ is a ${\gamma}-$entropy sub-solution with ${\gamma} = 1$.

Now we shall prove that $(n, c, u)$ as in Lemma 4.1 is a ${\gamma}-$entropy super-solution.

Lemma 4.4. Let $q\in\left({\frac{5}{3}, 2} \right)$ and $(n, c, u)$ be the limit functions and vector field in Lemma 4.1. Then for any fixed ${\gamma}\in\left({0, \frac{4q-5}{5}} \right)$, $n$ is a ${\gamma}-$entropy supersolution of $(1.1)–(1.3)$.

Proof. Let $0 \leqslant {\varphi}\in C^\infty_0(\overline{ {\Omega}}\times[0, \infty))$ be arbitralily. Testing the first equation in (2.1) by ${\gamma} n_ {\epsilon}^{ {\gamma}-1} {\varphi}$ and integrating by parts, we have

for all ${\epsilon}\in(0, 1)$. Since ${\gamma}\in(0, 1)$, we obtaing the strong convergence $n_ {\epsilon}^ {\gamma} {\rightarrow }\, n^ {\gamma}$ in $L^p_{loc}({\Omega}\times(0, \infty))$ for $p\in(1, q)$ due to (4.3) which follows

as ${\epsilon}\searrow 0$. Furthermore, referring to (4.20) and (4.21) we have

as ${\epsilon}\searrow 0$. As $n_ {\epsilon}^{q+ {\gamma}-1}$ is bounded in $L^k_{loc}({\Omega}\times(0, \infty))$ for $k = \frac{q}{q+ {\gamma}-1} >1$, uniformly in ${\epsilon}$, the weak convergence $n_ {\epsilon}^{q+ {\gamma}-1}\rightharpoonup n^{q+ {\gamma}-1}$ in $L^k_{loc}({\Omega}\times(0, \infty))$ holds. Thus, we have

as ${\epsilon}\searrow0$. Since $\frac{5+q}{5q}+\frac{ {\gamma}}{q} < 1$, it follows from (4.3) and (4.7) that

as ${\epsilon}\searrow0$. For the regularizing term, we note that from Hölder inequality and (3.2)

for all ${\epsilon}\in(0, 1)$. Hence, we have

as ${\epsilon}\searrow0$. Finally, from (4.4) and the lower semicontinuity of the seminorm $\left\Vert {\cdot} \right\Vert$ defined by $\left\Vert {f} \right\Vert: = (\int\limits_{0}^\infty\int\limits_{\Omega} f^2 {\varphi})^\frac{1}{2}$ with respect to weak convergence, we obtain

Therefore, collecting (4.24)–(4.29) proves that $n$ is a ${\gamma}-$entropy super-solution of (1.1)–(1.3).

Proof of Theorem 1.1. This is the combination of Lemma 4.2, Lemma 4.3 and Lemma 4.4.

5.   Asymptotic behavior

The following Lemma is elementary, but for clarity, we give its detail.

Lemma 5.1. Let $a > 1$ and $f\in L^1([0, \infty))$. Suppose there is $t_0 > 0$ such that $f(t) \leqslant Nt^{-a}$ for sufficiently large $t \geqslant t_0$. Assume further that a non-negative measurable function $y(t)$ satisfies

Then, $y(t) \leqslant Ct^{-a}$ for sufficiently large $t$.

Proof. Firstly we note that $y(t)$ is bounded uniformly in time. Then, using the integrating factor, we have for $t \geqslant t_0$

which yields, using integration by parts,

Proof of Theorem 1.2. $\bullet$ (The case $\rho = 0$) Noting that $\rho = 0$, we integrate the equation for $n_ {\epsilon}$ in (2.1) over $\Omega$ to get

A standard argument of ODE implies that

Next, integrating the equation of $c_ {\epsilon}$, it follows that for all $t>0$,

Let $g(t) = \int\limits_{\Omega} n_ {\epsilon}(\cdot, t) {\, \mathrm{d}} x$. Then, since $\frac{1}{q-1} > 1$, we observe that $g\in L^{1}([0, \infty))$, and thus, via Lemma 5.1, it follows that

On the other hand, putting $m = 3q-2$ and testing the equation for $c_ {\epsilon}$ in (2.1) by $c^{m-1}$, we get

Since $m = 3q-2$, we observe that

Let $h(t) = (1+t)^{-2}\left\Vert {n_ {\epsilon}(t)} \right\Vert_{q}^q$. Then, it is direct that $h\in L^1((0, \infty))$. Setting $Z(t) = \int\limits_{\Omega} c_ {\epsilon}^m(\cdot, t) {\, \mathrm{d}} x$, we have $Z'(t)+Z(t) \leqslant h(t)$, which yields

which implies that

Noting that $Z(t) \leqslant C$ for all $t>0$, we have

Hence, interpolation gives

where $1 \leqslant l \leqslant 3q-2$ and $\theta = \frac{(l-1)(3q-2)}{3l(q-1)}$. On the other hand, in case that $3q-2 \leqslant l \leqslant \frac{3q}{5-2q}$, interpolation gives

where $\theta_1 = \frac{(3q-(5-2q)k)(3q-2)}{2k(3q-5)(q-1)}$. Finally, recalling (3.14) and (3.15), we have

where we used

We set $h(t) = \left\Vert {n_ {\epsilon}(t)} \right\Vert^{\frac{5q-6}{3(q-1)}}_{1}\left\Vert {n_ {\epsilon}(t)} \right\Vert^{\frac{q}{3(q-1)}}_{q} \leqslant C(1+t)^{-\frac{5q-6}{3(q-1)^2}}\left\Vert {n_ {\epsilon}(t)} \right\Vert^{\frac{q}{3(q-1)}}_{q}$. We note that $h\in L^1((0, \infty))$, since $n_ {\epsilon}\in L^q({\Omega}\times(0, \infty))$ and

Using the Poincaré inequality, it follows that

Since $h$ is in $L^1$, we have $\left\Vert {u_ {\epsilon}(\cdot, t)} \right\Vert_{2} \leqslant C$ for all $t$. In addition, we obtain, for sufficiently large $t$,

Indeed, setting $z(t): = \left\Vert {u_ {\epsilon}(t)} \right\Vert^2_{2}$, it leads that

$\bullet$ (The case $\rho < 0$) Firstly, we integrate the equation for $n_ {\epsilon}$ over $\Omega$ to get

which directly yields

where m is as in Lemma 3.1. Next, again integrating the equation for $c_ {\epsilon}$ over $\Omega$ and letting $z(t): = \int_\Omega c_ {\epsilon}(\cdot, t) {\, \mathrm{d}} x$, it follows that

which leads that for all $t>0$,

where $C = \max\left\{ {m, \int_\Omega c_0} \right\}$. Thus, we have

where $\rho_* = \min\left\{ {-\rho, 1} \right\}>0$. Using the interpolation inequality, (3.5) and (5.3), we obtain for $1 \leqslant l \leqslant \frac{3q}{5-2q}$,

Lastly, we recall the inequality (5.1):

Here $h(t) = \left\Vert {n_ {\epsilon}} \right\Vert_1^\frac{5q-6}{3(q-1)}\left\Vert {n_ {\epsilon}} \right\Vert_q^\frac{q}{3(q-1)} \leqslant C_3e^{-\delta t}\left\Vert {n_ {\epsilon}(t)} \right\Vert_q^\frac{q}{3(q-1)}$ with $\delta = -\frac{5q-6}{3(q-1)}\rho>0$ and $C_* = \frac{C_p}{2}>0$, where $C_p$ is the constant appeared in the Poincaré inequality. Letting $z(t): = \left\Vert {u_ {\epsilon}(t)} \right\Vert_2^2$, we have

where $\delta_* = \frac{1}{2}\min\left\{ {C_*, \delta} \right\}$. In both cases $\rho = 0$ and $\rho < 0$, we finally get the estimates for $(n, c, u)$ in Theorem 1.2 by passing ${\epsilon}$ to the limit via the Fatou's Lemma which is guaranteed by (4.1), (4.5) and (4.8).

Acknowledgements

K. Kang is partially supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. D. Kim is supported by NRF-2019R1A2C1084685.

Conflict of interest

The authors declare no conflict of interest.